Singularities of the susceptibility of an SRB measure in the presence of stable-unstable tangencies

TL;DR

This paper analyzes how stable-unstable tangencies in dynamical systems affect the susceptibility function of SRB measures, revealing that the Hausdorff dimension of the measure influences the location of singularities and the potential for differentiability.

Contribution

It provides a nonrigorous analysis linking stable-unstable tangencies to singularities in the susceptibility function based on the Hausdorff dimension of the SRB measure.

Findings

Singularities occur at different locations depending on the Hausdorff dimension d.

If d<1/2, singularities are inside the unit circle, affecting convergence.

If d>1/2, singularities are outside the unit circle, allowing potential differentiation at z=1.

Abstract

Let $\rho$ be an SRB (or "physical"), measure for the discrete time evolution given by a map $f$, and let $\rho(A)$ denote the expectation value of a smooth function $A$. If $f$ depends on a parameter, the derivative $\delta\rho(A)$ of $\rho(A)$ with respect to the parameter is formally given by the value of the so-called susceptibility function $\Psi(z)$ at $z=1$. When $f$ is a uniformly hyperbolic diffeomorphism, it has been proved that the power series $\Psi(z)$ has a radius of convergence $r(\Psi)>1$, and that $\delta\rho(A)=\Psi(1)$, but it is known that $r(\Psi)<1$ in some other cases. One reason why $f$ may fail to be uniformly hyperbolic is if there are tangencies between the stable and unstable manifolds for $(f,\rho)$. The present paper gives a crude, nonrigorous, analysis of this situation in terms of the Hausdorff dimension $d$ of $\rho$ in the stable direction. We find that the tangencies produce singularities of $\Psi(z)$ for $|z|<1$ if $d<1/2$, but only for $|z|>1$ if $d>1/2$. In particular, if $d>1/2$ we may hope that $\Psi(1)$ makes sense, and the derivative $\delta\rho(A)=\Psi(1)$ has thus a chance to be defined